Introduction
Self-focusing of laser beams in plasmas (Chiao et al., Reference Chiao, Garmire and Townes1964; Tabak et al., Reference Tabak, Hammer, Glinsky, Kruer, Wilks, Woodworth, Campbell, Perry and Mason1994; Gill et al., Reference Gill, Saini, Kaul and Singh2004) is one of the most interesting phenomena in the field of research for several decades due to its various applications, like high harmonic generation (Ganeev et al., Reference Ganeev, Toşa, Kovács, Suzuki, Yoneya and Kuroda2015; Vij et al., Reference Vij, Gill and Aggarwal2016a, Reference Vij, Kant and Aggarwal2016b), laser-driven inertial confinement fusion (Hora, Reference Hora2007; Winterberg, Reference Winterberg2008), laser-based plasma acceleration (Sari et al., Reference Sari, Osman, Doolan, Ghoranneviss, Hora, Hopfl, Benstetter and Hantehzadehi2005; Niu et al., Reference Niu, He, Qiao and Zhou2008; Jha et al., Reference Jha, Saroch and Mishra2011, Reference Jha, Saroch and Mishra2013; Rajeev et al., Reference Rajeev, Madhu Trivikram, Rishad, Narayanan, Krishnakumar and Krishnamurthy2013), and generation of x-rays (Arora et al., Reference Arora, Naik, Chakera, Bagchi, Tayyab and Gupta2014).
The main thrust of the theoretical and experimental investigations on self-focusing of a laser beam has been directed toward the study of the propagation characteristics of a Gaussian beam (Akhmanov et al., Reference Akhmanov, Sukhorov and Khokhlov1968; Sodha et al., Reference Sodha, Ghatak and Tripathi1974, Reference Sodha, Ghatak and Tripathi1976; Sharma et al., Reference Sharma, Prakash, Verma and Sodha2003, Reference Sharma, Verma and Sodha2004; Singh et al., Reference Singh, Aggarwal and Gill2009; Aggarwal et al., Reference Aggarwal, Kumar and Kant2016). Subsequently, a few studies have been made on self-focusing of super Gaussian beams (Gill et al., Reference Gill, Mahajan, Kaur and Gupta2012), cosh-Gaussian beams (Patil et al., Reference Patil, Takale, Navare and Dongare2008, Reference Patil, Takale, Navare, Fulari and Dongare2012; Gill et al., Reference Gill, Kaur and Mahajan2011a, Reference Gill, Mahajan and Kaur2011b; Aggarwal et al., Reference Aggarwal, Vij and Kant2014; Vhanmore et al., Reference Vhanmore, Patil, Valkunde, Urunkar, Gavade and Takale2017), Hermite cosh-Gaussian beams (Patil et al., Reference Patil, Takale, Navare, Fulari and Dongare2007, Reference Patil, Takale, Navare and Dongare2010; Ghotra and Kant, Reference Ghotra and Kant2016; Kaur et al., Reference Kaur, Kaur, Kaur and Gill2017; Valkunde et.al., Reference Valkunde, Patil, Vhanmore, Urunkar, Gavade and Takale2018a, Reference Valkunde, Patil, Vhanmore, Urunkar, Gavade, Takale and Fulari2018b), dark-hollow Gaussian beams (Sodha et al., Reference Sodha, Mishra and Misra2009; Gill et al., Reference Gill, Mahajan and Kaur2010), quadruple Gaussian beams (Aggarwal et al., Reference Aggarwal, Vij and Kant2015a, Reference Aggarwal, Vij and Kant2015b; Vij et al., Reference Vij, Gill and Aggarwal2016a, Reference Vij, Kant and Aggarwal2016b), elliptic Gaussian beams (Saini and Gill, Reference Saini and Gill2006; Singh et al., Reference Singh, Aggarwal and Gill2008), q-Gaussian beams (Valkunde et.al., Reference Valkunde, Patil, Vhanmore, Urunkar, Gavade and Takale2018a, Reference Valkunde, Patil, Vhanmore, Urunkar, Gavade, Takale and Fulari2018b; Vhanmore et al., Reference Vhanmore, Patil, Valkunde, Urunkar, Gavade, Takale and Gupta2018). Recently, the propagation of Gaussian laser beam in three distinct regimes has been studied by Sharma et al. (Reference Sharma, Prakash, Verma and Sodha2003, Reference Sharma, Kourakis and Sodha2008). Such propagation regimes include steady divergence, oscillatory divergence, and self-focusing of laser beams.
In recent years, considerable interest has been evinced toward the production and propagation of decentered Gaussian beams, usually known as cosh-Gaussian beams on account of their wide and attractive applications in complex optical systems (Lu et al., Reference Lu, Ma and Zhang1999; Lu and Luo, Reference Lu and Luo2000) and turbulent atmosphere (Chu Reference Chu2007; Chu et al., Reference Chu, Ni and Zhou2007). The propagation properties of cosh-Gaussian laser beams have important technological issues, since these beams possesses high power in comparison to that of a Gaussian beam (Konar et al., Reference Konar, Mishra and Jana2007). The self-focusing of cosh-Gaussian laser beam passing through different plasma media have been studied (Sodha et al., Reference Sodha, Mishra and Agarwal2007; Patil et al., Reference Patil, Takale, Navare and Dongare2008, Reference Patil, Takale, Navare, Fulari and Dongare2012; Gill et al., Reference Gill, Kaur and Mahajan2011a, Reference Gill, Mahajan and Kaur2011b; Aggarwal et al., Reference Aggarwal, Vij and Kant2014; Nanda and Kant, Reference Nanda and Kant2014; Vhanmore et al., Reference Vhanmore, Patil, Valkunde, Urunkar, Gavade and Takale2017). Moreover, all the above references discuss the effect of decentered parameter on self-focusing. However, in the present attempt, authors have carried out an exploratory study of critical beam power by employing a parabolic equation approach under Wentzel–Kramers–Brillouin (WKB) and paraxial approximations.
The organization of the paper is as follows: “Basic formulation” section gives the evolution of beam-width parameter equation. Discussion of results in the context of self-focusing of cosh-Gaussian laser beams is elaborated in “Results and discussion” section. Finally, “Conclusion” section involves overall conclusions drawn from the present study.
Basic formulation
Let us consider the propagation of cosh-Gaussian laser beams through collisionless magnetized plasma along the z direction, which is the direction of static magnetic field B 0. The electric field of the laser beam propagating in either of the two modes, that is, extraordinary and ordinary can be written as,

where $k_ \pm = {\rm \omega} /c\sqrt {{\rm \varepsilon} _{0 \pm}} $ is the propagation constant of the wave. Here ε0± is the linear part of plasma dielectric constant and c is the speed of light in vacuum. The effective dielectric constant of magnetized plasma can be written as,

where ${\rm \varepsilon} _{0 \pm} = 1 - {\rm \omega} _{\rm p}^2 /{\rm \omega} ({\rm \omega} \mp {\rm \omega} _{\rm c})$ is the linear part of dielectric constant with ωp = (4πN 0e 2/m)1/2 as a plasma oscillation frequency and ωc = eB 0/mc as a cyclotron frequency. Here, e and m are the electronic charge and rest mass, respectively.
The second term in [Eq. (2)], the intensity-dependent part of dielectric constant for a collisionless magnetized plasma, is given by

In the light of Maxwell's equations, the general form of wave equation governing the propagation of laser beam is given as,

where ${\rm \delta} _ \pm = 1/2(1 \mp {\rm \varepsilon} _{0 \pm} /{\rm \varepsilon} _{0zz})$ and
${\rm \varepsilon} _{0zz} = 1 - {\rm \omega} _{\rm p}^2 /{\rm \omega} ^2.$ The amplitude of electric field E ± in laser beam is given by [Eq. (1)] which satisfies [Eq. (4)]. Under slowly varying envelope approximation, the evolution of electric field envelope in collisionless magnetized plasma can be written as,

In WKB approximation, one can neglect ∂2E ±/∂z 2 from [Eq. (4)]. The complex amplitude of electric vector may be expressed as,

where A ±(r, z) and S ±(r, z) are the real functions of r and z, S ± is eikonal. Substituting [Eq. (6)] in [Eq. (5)] and separating into real and imaginary parts,

and

The solution of [Eqs. (7) and (8)] for cosh-Gaussian laser beam can be written as



where $E_{0 \pm} ^2 $ is an initial laser intensity and f ± is the dimensionless beam-width parameter for extraordinary and ordinary modes, respectively. The
${\rm \beta} _ \pm ^{ - 1} $ is the radius of the curvature of wave front, φ±(z) is the phase shift, and f ± is the beam-width parameter, which is the measure of both axial intensity and width of the beam. Substituting [Eqs. (9a)–(9c)] in [Eq. (7)] and equating coefficients of r 2 on both sides we get

where





The quantity p ± which is proportional to $E_{0 \pm} ^2 $ represents the beam power. The beam-width parameters
$f_ \pm ^{} $ are the functions of
${\rm \xi} _ \pm ^{} $, with
${\rm \xi} _ \pm = z/k_ \pm r_0^2 $ as the normalized propagation distance.
Results and discussion
Equation (10) is a second-order, non-linear differential equation which represents variation of beam-width parameter f ± with normalized distance of propagation ξ ±. The first term on the right-hand side of [Eq. (10)] corresponds to the diffraction divergence of the beam and the second term corresponds to the convergence resulting from the ponderomotive non-linearity.
By subjecting [Eq. (10)] under critical condition (f ± = 1, ξ ± = 0), the general power of laser beam p ± and beam radius ${\rm \rho} _ \pm ^{} $ are replaced by critical power p 0± and critical beam radius
${\rm \rho} _{0 \pm} ^{} $. Therefore, right-hand side of [Eq. (10)] takes the form

Henceforth following investigation is restricted only for extraordinary mode.

The condition that the critical beam power should be always finite and positive leads to the numerical interval of decentered parameter as 0 ≤ b ≤ 0.9634. By reducing defining equations for δ+, γ + and ${\rm \rho} _{0 +} ^{} $ numerically with the help of values N 0 = 1 × 1018cm−3, ω = 1.776 × 1015rad/s, B 0 = 106gauss, r = 20 × 10−4cm and by defining the values of decentered parameter b = 0.00, 0.45, 0.90. [Eq. (12)] depends upon critical beam power.
To explore the effect of critical beam power right at the beginning one has to pay little attention to the plot shown in Figure 1. From the plot, it is clear that at the beginning, two values of critical beam powers p 0+lower and p 0+upper are possible.

Fig. 1. Variation of F(p 0+) with p 0+.
The function F(p 0+) has minimum value at p 0+ = 0.98, which is a common value for all decentered parameters ranging from 0.00 to 0.90. Hence, the value of p 0+ <0.98 and p 0+ >0.98 can be considered as p 0+lower and p 0+upper, respectively.
The plot shown in Figure 1 can be studied for three distinct conditions stated below and the simple analytical approach leads to the following limits for critical beam power.
For self-trapping:

For defocusing:

For self-focusing:

In the present analysis, authors are interested only in the self-focusing region in Figure 1.
From Figure 1, it is also seen that for above limits of critical beam powers for the condition of self-focusing of laser beams, three curves for three distinct values of decentered parameters intersect each other at three distinct points close to p 0+lower and p 0+upper as shown in insets of Figure 1.
In the analytical investigation, the intersecting points are called as turning points, which affect the propagation of laser beams through collisionless magnetized plasma. The turning points are as follow:

The above study shows that preconditioning of a critical beam power in the beginning of propagation of laser beam can determine propagation dynamics effectively.
In the lower region, (p 0+ <0.98) of Figure 2a, 2b, which is the present region of interest, gives variation of beam-width parameter f + against normalized propagation distance ξ +. In Figure 2a, 2b, strong self-focusing is observed. However, in Figure 2a, periodicity in self-focusing length is observed over larger interval of ξ + as compared with periodicity in self-focusing length observed in Figure 2b over a given range of ξ +. Additionally, with increase in the values of decentered parameter, the effect of decentered parameter b on self-focusing length in Figure 2a is exactly opposite to that in Figure 2b. From the insets of Figure 2a, 2b, it is evident that the rate of self-focusing is exactly opposite with increase in the values of decentered parameter b.

Fig. 2. (a) Variation of beam-width parameter f + with dimensionless propagation distance ξ + before turning point for p 0+lower = 0.0035. (b) Variation of beam-width parameter f + with dimensionless propagation distance ξ + after turning point for p 0+lower = 0.15.
In the upper region, (p 0+ >0.98) of Figure 3a, 3b, which is the subsequent region of interest, gives variation of beam-width parameter f + against normalized propagation distance ξ +. In Figure 3a, 3b, weak self-focusing is observed and the effect of decentered parameter b on self-focusing length is same, that is, with increase in the values of decentered parameter self-focusing, the length increases. Moreover, from Figure 3a, it is observed that with increase in the values of decentered parameter b, the rate of self-focusing is exactly opposite to that observed in Figure 3b, which is also evident from the insets of Figure 3a, 3b.

Fig. 3. (a) Variation of beam-width parameter f + with dimensionless propagation distance ξ + before turning points for p 0+upper = 5.5. (b) Variation of beam-width parameter f + with dimensionless propagation distance ξ + after turning point for p 0+upper = 7.2.
Conclusion
In conclusion, the numerical intervals of critical beam power for the non-linear phenomena such as self-trapping, self-focusing, and defocusing of cosh-Gaussian laser beams are studied. Following, important conclusions are obtained for collisionless magnetized plasma from the present investigation.
• Two values of critical beam power (p 0+lower, p 0+upper) are possible for each specified value of decentered parameter under self-trapping condition of cosh-Gaussian laser beams.
• In the lower region, exactly opposite trends of self-focusing before and after turning points are observed with increase in the values of decentered parameter. It is to be noted that in the lower region, self-focusing length before turning points and after turning points shows exactly opposite trends.
• In the lower and upper regions, rate of self-focusing is exactly opposite with increase in the values of decentered parameter before and after turning points.